MATH 614
Dynamical Systems and Chaos
Lecture 17a:
Morse-Smale diffeomorphisms.
Maps of the circle
T : S 1 → S 1,
T an orientation-preserving homeomorphism.
Structurally stable maps of the circle
Definition. An orientation-preserving diffeomorphism
f : S 1 → S 1 is Morse-Smale if it has rational rotation
number and all of its periodic points are hyperbolic.
If ρ(f ) = m/n, a reduced fraction, then all periodic points of
f have period n. Hence the only periodic points of f n are
fixed points, alternately sinks and sources around the circle.
Theorem A Morse-Smale diffeomorphism of the circle is
C 1 -structurally stable.
Theorem (The Closing Lemma) Suppose f is a
C r -diffeomorphism of S 1 with an irrational rotation number.
Then for any ε > 0 there exists a diffeomorphism
g : S 1 → S 1 with a rational rotation number such that f and
g are C r -ε close.
Theorem (Kupka-Smale) For any orientation-preserving
C r -diffeomorphism f of S 1 and any ε > 0 there exists a
Morse-Smale diffeomorphism that is C r -ε close to f .